Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 3x) dx.

Find the value of ∫ from 0 to 1 of (x^3 - 3x^2 + 3x) dx.

Correct Answer: 1/4

Step 1: Identify the integral you need to solve: ∫ from 0 to 1 of (x^3 - 3x^2 + 3x) dx.
Step 2: Find the antiderivative of the function (x^3 - 3x^2 + 3x).
Step 3: The antiderivative is calculated as follows: For x^3, the antiderivative is x^4/4; for -3x^2, it is -x^3; and for 3x, it is (3/2)x^2.
Step 4: Combine the antiderivatives: The complete antiderivative is (x^4/4 - x^3 + (3/2)x^2).
Step 5: Evaluate the antiderivative from 0 to 1: Substitute 1 into the antiderivative: (1^4/4 - 1^3 + (3/2)(1^2)) = (1/4 - 1 + 3/2).
Step 6: Simplify the expression: 1/4 - 1 + 3/2 = 1/4 - 4/4 + 6/4 = (1 - 4 + 6)/4 = 3/4.
Step 7: The final answer is 3/4.

Definite Integral – The question tests the ability to evaluate a definite integral, which involves finding the area under the curve of the function between specified limits.
Polynomial Integration – The integral involves integrating a polynomial function, which requires applying the power rule for integration.
Fundamental Theorem of Calculus – The question assesses understanding of the Fundamental Theorem of Calculus, which connects differentiation and integration.